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Mirrors > Home > MPE Home > Th. List > f1ocnvd | Structured version Visualization version GIF version |
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
f1od.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
f1od.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) |
f1od.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) |
f1od.4 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
Ref | Expression |
---|---|
f1ocnvd | ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1od.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) | |
2 | 1 | ralrimiva 3140 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊) |
3 | f1od.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | 3 | fnmpt 6645 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊 → 𝐹 Fn 𝐴) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
6 | f1od.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) | |
7 | 6 | ralrimiva 3140 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋) |
8 | eqid 2733 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑦 ∈ 𝐵 ↦ 𝐷) | |
9 | 8 | fnmpt 6645 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
11 | f1od.4 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | |
12 | 11 | opabbidv 5175 | . . . . . 6 ⊢ (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}) |
13 | df-mpt 5193 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
14 | 3, 13 | eqtri 2761 | . . . . . . . 8 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
15 | 14 | cnveqi 5834 | . . . . . . 7 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
16 | cnvopab 6095 | . . . . . . 7 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
17 | 15, 16 | eqtri 2761 | . . . . . 6 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
18 | df-mpt 5193 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)} | |
19 | 12, 17, 18 | 3eqtr4g 2798 | . . . . 5 ⊢ (𝜑 → ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)) |
20 | 19 | fneq1d 6599 | . . . 4 ⊢ (𝜑 → (◡𝐹 Fn 𝐵 ↔ (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)) |
21 | 10, 20 | mpbird 257 | . . 3 ⊢ (𝜑 → ◡𝐹 Fn 𝐵) |
22 | dff1o4 6796 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | |
23 | 5, 21, 22 | sylanbrc 584 | . 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
24 | 23, 19 | jca 513 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 {copab 5171 ↦ cmpt 5192 ◡ccnv 5636 Fn wfn 6495 –1-1-onto→wf1o 6499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 |
This theorem is referenced by: f1od 7609 f1ocnv2d 7610 pw2f1ocnv 41408 |
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